That is an excellent point and an excellent observation!
You are completely correct.
Anyway, it is customary to say PBUH, etc, after the name of the Prophet (S) as a gesture of respect, and also because the Qur'an says to send blessings upon the Prophet (S). After all, there is no limit to Allah's blessings!
When we pray for someone to be blessed, it also elevates us, because it is a selfless act of giving. And it attracts Allah's blessings to ourselves.
However, I do feel that this type of usage can become ritualistic and dry if it is just done out of habit, so it is good to think about.
Also, I am of the view that it is not really necessary to do frequently in English because it looks/sounds weird and omitting it, in the custom of the English language, does not imply disrespect to the Prophet. Especially when speaking to or writing things for people who are not Muslims, as it is very weird to see PBUH all over the place. But other people may have other views.